Extending an idea of Suppakitpaisarn, Edahiro and Imai, a dynamic programming approach for computing digital expansions of minimal weight is transformed into an asymptotic analysis of minimal weight expansions. The minimal weight of an optimal expansion of a random input of length $\ell$ is shown to be asymptotically normally distributed under suitable conditions. After discussing the general framework, we focus on expansions to the base of $\tau$, where $\tau$ is a root of the polynomial $X^2- \mu X + 2$ for $\mu \in \{ \pm 1\}$. As the Frobenius endomorphism on a binary Koblitz curve fulfils the same equation, digit expansions to the base of this $\tau$ can be used for scalar multiplication and linear combination in elliptic curve cryptosystems over these curves.